In her book What's Math Got to Do With It?, Jo Boaler recounted her first contact with the math wars. A group of parents at a local school had organized against the adoption of reform textbooks, and were telling students that if they took the classes with the new books, they wouldn't be eligible for college. Apparently the parents had called admissions offices and asked a question like, "Would you accept a student who had not taken any math in high school but had just talked about math?" (Boaler, 2008, p. 33). Of course the colleges said no, and from that question and response parents based a claim about reform texts and college admission.

If we try to put angry politics aside for a moment, where did the parents in Boaler's story get the idea that reform math was all about talking about math? Certainly that thought was not a total fabrication. In fact, the learning theories that influenced much of the reform math movement led teachers and researchers to think about how to structure classrooms in ways that maximized learning, and that led to an attention to classroom discourse -- the teacher-student and student-student speaking and writing that happens in classrooms. By studying classroom discourse, we can gain key insights about what and how math is learned, and how our experiences and our environment affect that learning. (This, I believe, is not unique to reform classrooms -- all good math teachers and students pay careful attention to how we talk about and otherwise communicate mathematics.)

Megan Staples, a former advisee of Jo Boaler, is an assistant professor at the University of Connecticut specializing in mathematics education and classroom discourse. In her article Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom (2007), Staples takes on the challenges of being the teacher and supporting students in "collaborative inquiry." Instead of simply viewing collaboration as working in groups, or cooperating to complete a task, Staples says collaboration, "implies a joint production of ideas, where students offer their thoughts, attend and respond to each other's ideas, and generate shared meaning or understanding through their joint efforts" (p. 162). As for inquiry, Staples sees it as a way of "engaging with and making sense of the world" (p. 163), and applies the term to "both inquiry into mathematics and inquiry with mathematics" (p. 163). Finally, and perhaps most importantly, is how Staples defines learning mathematics itself. Using a situative perspective, Staples sees mathematics as a cultural practice, where "learning results from, and is evidenced by, student participation in both standard and disciplinary practices (e.g., justifying, representing algebraically) and an array of other practices of mathematical communities (e.g., questioning, communicating, informal reasoning)" (p. 163). With these conceptions of collaboration, inquiry, and learning mathematics in mind, Staples conducted a year-long observation and analysis of Ms. Nelson, a veteran, award-winning teacher known for her dedication to reform mathematics principles. The class Staples analyzed was called "Math A," a lower-track 9th grade class for students with a history of low performance in traditional mathematics classrooms.

Staples's analysis yielded two models for teaching that support student participation in class discussion. The first model describes the role of the teacher in a whole-class discussion, while the second describes how Ms. Nelson increases the class's ability to collaborate over time. I'll present Staples's findings in outline form, with attention to specific recommendations for teachers wishing to support collaborative inquiry in their own classrooms. (Be patient -- the original article is 57 pages long, after all.)

Model 1: The teacher's role in supporting whole-class collaborative inquiry

Supporting students in making contributions

Eliciting student ideas -- Instead of just asking questions like, "Why?" or "How do you know?," Ms. Nelson presses students to share with comments like, "Come on, I'm really interested, come on, you can do it" (p. 175), gave students adequate time to formulate explanations, and offered participation points as a reward for contributing ideas.

Scaffolding the production of student ideas -- Ms. Nelson helps direct struggling students to use multiple representations and models, such as number lines, graphs, diagrams, etc. The key, says Staples, is to provide structure for the mathematics without constraining how the students will work out the mathematics (p. 178).

Creating contributions - Ms. Nelson treated incomplete and incorrect contributions by students the same as correct ideas, saying things like, "Remember the idea is to go up and give us some good discussion...that helps the class move along regardless of whether it's right or wrong, it enables us to have good discussion" (pp. 178-179).

Establishing and monitoring a common ground

Creating a shared context -- Ms. Nelson focused on creating shared contexts among students. This was accomplished by repeating of student statements and encouraging students to record and share their representations and ideas on the board.

Maintaining continuity over time -- Ms. Nelson emphasized a sense of purpose when asking students to contribute. She directed students with phrases like, "Come up [to the board] please. Ron says that there are more diagonal lines. That Oscar didn't put enough in" (p. 181). Ms. Nelson also gave students time to understand and add clarity to other students' ideas before introducing new ideas.

Coordinating the collective -- Ms. Nelson actively positions students to respond to each other. When a student, Jay, had difficulty explaining an idea and Ken raised his hand, Ms. Nelson asked, "OK, do you wanna explain some more Ken? Or do you have a question for Jay?" (p. 185).

Guiding the mathematics

Guiding high-level task implementation -- Ms. Nelson selected tasks that were difficult enough to invite collaboration, but guided students in ways that avoided unproductive exploration. This sometimes involved recounting the steps students had taken to reach their current thinking or requesting new representations of ideas. Either way, the focus was on how the students were thinking about the problem, and not just giving hints for the next step.

Guiding with a map of students' algebra learning -- This is where Ms. Nelson showed her experience with mathematics and the learning of mathematics, knowing the "pressure points" (p. 190) where students needed to pay particular attention to the structure of the mathematics.

Guiding by following: "going with the kids" -- Ms. Nelson showed a willingness to let go and follow students' thinking and be flexible with the intended destination of the lesson.

Model 2: The development of a community of collaborative learners

The development of practices over time -- High school is a difficult time to introduce collaborative inquiry because students have longer histories with traditional mathematics and because classes meet for a limited time each day. Expectations need to be made explicit and modeled for students.

The model -- Developing community is an iterative process involving tasks or strategies that Staples calls "cycle starters" (p. 196) that spur student participation, which elicits negotiation of meanings, which leads to student interpretations and understandings. From there the cycle can repeat and improve.

Negotiation of meanings -- For example, early in the year students, when asked to explain, automatically assume they've given a wrong answer. Once this practice is established, students improve in the ways they respond to questions about their thinking.

Helping students make sense of practices -- Early in the year, Ms. Nelson would fill in commentary when students did work silently in front of the class, saying things like, "He is noticing a pattern over here" (p. 198). This modeled the practice of thinking aloud for the class and emphasized the value of sharing one's thinking.

Providing evidence for the value for learning -- When students struggled, Ms. Nelson encouraged them to stop and express what it was they were struggling with. Making mistakes became an acceptable part of doing mathematics so long as they became opportunities to learn and correct misunderstandings.

Negotiation of the joint enterprise -- Ms. Nelson explained early in the year that these students were doing to do mathematics differently than in the past, and that they didn't need the math dumbed-down just because they hadn't been successful before.

Cycle starters -- Ms. Nelson included not only engaging tasks, but helped create a vision of how that task could be accomplished, sometimes by describing the expected collaboration but also by showing a video of older students collaborating and discussing how they worked together.

Students' interpretations and understandings of practices -- From student interviews and surveys, Staples found that students interacted with each other during class either for social reasons or to make the class less boring. By the end of the year, students reported that working together helped them learn because of the opportunities to listen and respond to each others' ideas.

Transforming a community's repertoire -- Ms. Nelson's effort to transform the way it does mathematics was an ongoing effort that lasted the entire year. The effort is a negotiation, where as the class built new experiences together they could reflect on what was working and adjust their practices in future lessons.

In her discussion, Staples focuses on how teachers support collaborative inquiry while maintaining their role. It is certainly possible for a teacher to ask students to share ideas or report strategies, but it takes extra effort to build that common ground where students analyze and evaluate each other's ideas. Defining this common ground is difficult and it will be different in every classroom, but it is up to the teacher to develop and maintain it throughout the school year. Teachers also must be mindful of the mathematics, even when "going with the kids." It takes a skillful teacher to subtly push the mathematics while keeping the class collaborative. Lastly, the teacher must maintain a sense for a "long-term trajectory of student learning" (p. 210), although having such a sense does not necessarily support collaboration by itself. Staples's research is thorough and well-grounded in qualitative methodology, but it is not without its criticisms. I see critiques coming from two directions: from a more cognitive perspective, Staples doesn't attend much to individual student thinking, preferring to focus on social practices and classroom norms for participation. From a more purely sociocultural perspective, Staples doesn't account for how the influence of other, beyond-the-classroom cultures and community norms affect how students approach and understand mathematics. This is not to fault Staples, however -- she prefaced her findings with defining a situative perspective, and she maintained that perspective throughout. This just means that there are multiple ways of describing classroom communities and learning, and more work can be done to describe and build bridges across multiple perspectives.

References

Boaler, J. (2008). What’s math got to do with it? How parents and teachers can help children learn to love their least favorite subject (p. 273). New York, NY: Penguin Group.